A Simple form for the Two-Dimensional Q-function Suitable for Performance Evaluation of Communication Systems
Bibliographic record
Abstract
Using a geometrical approach inspired by the method of Craig for the one-dimensional Gaussian integral (one-dimensional Q-function, 1DQ) and similar to that of Simon and Divsalar, we propose a new form for the two-dimensional Gaussian integral (two-dimensional Q-function, 2DQ). Unlike that of Simon and Divsalar, our alternate form is applicable to any set of arguments and is not limited to nonnegative arguments. This rewrites the infinite-range double-integral expression in the naive form of 2DQ in terms of simple finite-range single integrals. The expressions given are simpler than those obtained by writing the 2DQ for negative arguments in terms of a sum of a number of 2DQ terms with nonnegative arguments. In addition, for a fixed required accuracy, the numerical complexity of the method is smaller than the rational function approximations of Donnelly.
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How this classification was reachedexpand
Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.004 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.001 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".